One-Sided Statistical Inference for a Multivariate Location Parameter

Transcription

1 One-Sided Statistical Inference for a Multivariate Location Parameter Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Michael Vock von Aarau und Wohlen AG Leiter der Arbeit: Prof. Dr. J. Hüsler Institut für mathematische Statistik und Versicherungslehre

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3 One-Sided Statistical Inference for a Multivariate Location Parameter Inauguraldissertation der Philosophisch-naturwissenschaftlichen Fakultät der Universität Bern vorgelegt von Michael Vock von Aarau und Wohlen AG Leiter der Arbeit: Prof. Dr. J. Hüsler Institut für mathematische Statistik und Versicherungslehre Von der Philosophisch-naturwissenschaftlichen Fakultät angenommen. Bern, 25. Januar 2007 Der Dekan: Prof. Dr. P. Messerli

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5 i Abstract In this thesis, we approach the problem of one-sided statistical tests and corresponding confidence regions for a multivariate location parameter in the onesample setting. By one-sided, we mean that these methods incorporate information on the direction of a deviation from the null hypothesis that shall be detected. A strong emphasis is on nonparametric methods. Particular attention is also paid to the distinction between methods based on simple and those based on composite null hypotheses. We propose several new test procedures and a graphical method for the comparison of different tests. This graphical method is useful for the assessment of the appropriateness of a test for a specific composite null hypothesis. Finally, we use a highly flexible definition of confidence regions to derive results about their properties from those of the corresponding tests. Particularly, we obtain results about the shape of such regions. Third printing, September 2007

6 ii Contents 1 Introduction and Outline Historical Background Outline One-Sided Multivariate Location Hypotheses Basic Multivariate One-Sided Alternatives Direction Alternatives Sector and Cone Alternatives Simple vs. Composite Null Hypotheses Hypotheses for Non-Inferiority Problems Translations of the Hypotheses Two- and Multi-Sample Problems Properties of Tests Invariance Properties Affine Transformations of p-variate Data Invariance of Tests More General Transformation Classes Transformations and One-Sided Tests Unbiasedness Cone Order Monotonicity Assumptions on the Underlying Distribution One-Sided Tests from the Literature One-Sample Problem Two-Sample Problem Multi-Sample Problem Monotone Trend Problem Combination of Univariate Tests Related Problems

7 Contents iii 5 Union-Intersection and Intersection-Union Tests Componentwise Procedures for the (2 p 1)-Orthant Alternative Componentwise Procedures for the One-Orthant Alternative Union-Intersection Tests Intersection-Union Tests Binomial Tests Half-Space Cone/Opposite Cone Estimated Cone Probability Spatial Sign and Rank Tests Spatial Sign Test for an Unrestricted Alternative Modification for a Direction Alternative Adaptation to a Sector Alternative Spatial Signed Rank Test Tests Based on the Procedure by Randles (2000) Sign Test for an Unrestricted Alternative Modification for a Direction Alternative Adaptation to a Sector Alternative Hodges-Type Tests Hodges Bivariate Sign Test Sign Test by Larocque and Labarre (2004) Modification for a One-Orthant Alternative Extension to a Convex Cone Alternative Graphical Comparison Methods Acceptance and Rejection Regions Power at a Fixed Distance from the Origin Curve of Constant Power Comparison of the Proposed Tests Uncorrelated Normal Case Power at a Fixed Distance from the Origin Curve of Constant Power Effects of Correlation in the Normal Case Heavy Tails: Bivariate t(1) Exponential Radius Distribution

8 iv Contents 12 Confidence Regions Connection with Hypothesis Tests Shape of the Confidence Region for the Meta-Parameter γ Translations of the Acceptance Region in R p Bounds for the Border of the Confidence Region Bivariate Case Confidence Regions for ϑ Based on Simple Null Hypotheses Sharpened Confidence Regions for ϑ Example: Pulmonary Function Data Data Tests Confidence Regions Discussion and Outlook 93 Acknowledgements 95 A Symmetry Centers and Expectations 96 A.1 Interpretation of Symmetry Centers Using Expectations A.2 Direction of the Expected Spatial Sign B Underlying Results 101 B.1 Asymptotic Distribution of Random Vectors B.2 Inequalities for Binomial Probabilities B.3 Minimum of Two Projections of an N 2 (0, I 2 ) Random Vector B.4 Maximum of a Symmetric Binary Random Walk B.5 Convex Cones C Algorithm for the Estimation of the Curve of Constant Power 106 Bibliography 108 Index 115

9 1 Chapter 1 Introduction and Outline 1.1 Historical Background Elementary statistical textbooks provide many univariate data examples, and the statistical methods for their analysis are well-known, even to many nonstatisticians. However, practically arising scientific questions tend to be more complicated very often, more than one variable is of interest. It may be much more appropriate to analyze the information of several variables at once instead of doing separate analyses. An early example of such a multivariate method can be found in the article by Hotelling (1931), in which the T 2 statistic was proposed as a multivariate generalization of Student s t for tests about a location parameter. Early nonparametric multivariate location tests were proposed by Hodges (1955) and Blumen (1958) (both for the bivariate case). The topic of such nonparametric tests has been receiving considerable attention during the last years; examples are the articles by Brown and Hettmansperger (1987), Randles (1989, 2000), Möttönen and Oja (1995), as well as Larocque, Tardif, and van Eeden (2000). We will deal with one-sided statistical inference for a multivariate location parameter. By one-sided, we mean that these methods incorporate information on the direction of a deviation from the null hypothesis that shall be detected. (An alternative hypothesis corresponding to such a test will also be called restricted.) As we will see in Chapter 2, there are many more possibilities for the specification of such information in the multivariate case than in the univariate case. Lehmann (1952) discussed general issues in testing one type of one-sided hypotheses. Kudô (1963) and Perlman (1969) made early proposals for multivariate one-sided tests under the assumption of normality. Among the first nonparametric procedures for such problems (in the two-sample case) are those by Bhattacharyya and Johnson (1970) and Chatterjee and De (1972). (Many further nonparametric one-sided location tests will be mentioned in Chapter 4.)

10 2 1 Introduction and Outline Most of the publications emphasize the concept of statistical tests, and not the dual concept of confidence regions. While confidence regions are immediately implied by the formulation of a (non-randomized) test, the practical aspects of such methods for multivariate one-sided problems do not seem to have received much attention. 1.2 Outline We start with a chapter about different formulations of one-sided null and alternative hypotheses about a multivariate location parameter. Chapter 3 introduces several properties that may be desirable for the tests, e. g. invariance properties under transformation classes, but also validity under different assumptions on the distribution of the data. In Chapter 4, we give a review of multivariate tests with restricted alternatives from the literature. The following chapters are focused on one-sample tests for cone (especially convex cone) alternatives. We begin with tests based on componentwise methods and so-called union-intersection and intersection-union tests as a related, but more general approach (Chapter 5). In Chapter 6, several nonparametric tests based on the number of observations in the alternative parameter region are discussed. Chapters 7 through 9 present specific nonparametric multivariate location tests proposed in the literature and modifications of these tests for one-sided problems. We discuss two existing and one novel approach to graphical comparisons of different multivariate one-sided location tests in Chapter 10. We then apply our graphical approach to a selection of our own tests and of tests from the literature in Chapter 11. In Chapter 12, we examine the connection between hypothesis tests and confidence regions. We use this connection to derive shape properties for different types of one-sided confidence regions for multivariate location parameters. Finally, hypothesis tests and confidence regions are illustrated using an example from the literature in Chapter 13, and the main achievements of the thesis as well as possible extensions are discussed in Chapter 14.

11 3 Chapter 2 One-Sided Multivariate Location Hypotheses The usual p-variate one-sample location problem can (up to translations, see Section 2.4) be formulated as H 0 : ϑ = 0 vs. H 1 : ϑ 0, where ϑ is a location parameter in R p. (More general alternatives than this shift alternative will not be considered here.) The corresponding k-sample problem (k 2) is H 0 : ϑ 1 =... = ϑ k vs. H 1 : ϑ i ϑ j for some i,j {1,...,k}. In some cases, the possible range of parameters is only a subset of R p. As an example, some components may be restricted to be non-negative by theoretical considerations. Furthermore, before looking at the data, one may already have an idea of the direction of the deviation of the location parameter from the null value, or it may even be the aim of a study to show a deviation into a specific direction. In all these cases, conventional tests with unrestricted alternatives are inappropriate. Their power may be unacceptably low, and they do not allow for showing a deviation into a specified direction. Thus special tests are needed that are able to deal with restricted alternatives. (The above-mentioned, quite different reasons for the usage of restricted alternatives will be returned to in Section 2.2.) The usual restricted alternatives in the one-dimensional case are the wellknown one-sided alternatives. It is not obvious how the concept of one-sided alternatives should be generalized to the multivariate case, so that careful consideration seems worthwhile. Emphasis will be on the one-sample location problem.

12 4 2 One-Sided Multivariate Location Hypotheses 2.1 Basic Multivariate One-Sided Alternatives In the one-dimensional one-sample setting, there are (up to translations) only two possible one-sided location (shift) alternatives: H + 1 : ϑ > 0 H 1 : ϑ < 0, and where the (simple) null hypothesis is H 0 : ϑ = 0. The generalization to the multivariate case can be done in different ways. For the ease of formulation and visualization, we focus on the two-dimensional setting at first Direction Alternatives We can interpret the univariate one-sided alternatives as the (only) two directions from the origin on the real line. In the bivariate case, if ϑ = (ϑ 1,ϑ 2 ) T is the (now vector-valued) location parameter again, we can e. g. consider restricted alternatives of the following form: H ϕ 1 : ϑ,u > 0, where, denotes a scalar product (typically, the standard Euclidean scalar product) and u = (cosϕ, sin ϕ) T is the unit vector at the angle ϕ; cf. Figure 2.1. Thus a projection of ϑ is investigated. We call this type of alternative hypothesis a direction alternative. The generalization of the direction alternative approach to the general p- variate case is straightforward. 1 u ϕ Figure 2.1: Specification of a direction alternative.

13 2.1 Basic Multivariate One-Sided Alternatives Sector and Cone Alternatives Another possibility is to generalize the idea of partitioning the real line into the positive half-line R + and the negative half-line R. The simplest possibility in the bivariate case would be to consider half-planes, which is closely related to the approach using projections. It is perhaps more promising to use the more general concept of partitioning the plane into two sectors, i.e. to ask if the location parameter ϑ belongs to a specified (infinite) sector between the angles ϕ 1 and ϕ 2 (assuming, without loss of generality, ϕ 1 < ϕ 2 ); cf. Figure 2.2 (a). The directions can equivalently be given by the unit vectors u 1 and u 2 again. Formally, we can specify such a restricted alternative as H ϕ 1,ϕ 2 1 : ϑ 0; ψ [ϕ 1,ϕ 2 ] : cos ψ = ϑ 1 ϑ, sin ψ = ϑ 2 ϑ. (Sometimes, we will also use the open interval (ϕ 1,ϕ 2 ) instead of the closed interval.) We call this type of alternative a sector alternative. This approach includes the case of half-planes (cf. Figure 2.2 (b), with a boundary slightly different from that in the direction alternative above). It also includes the quadrant alternatives, which are of considerable interest in many applications and have been discussed in several articles (e. g. Kudô, 1963; Chinchilli and Sen, 1981; Chatterjee and De, 1972; Boyett and Shuster, 1977; Park, Na, and Desu, 2001). A typical example for a one-quadrant alternative (cf. Figure 2.2 (c)) is the comparison of treatments where a new treatment should be superior to an old one in two criterions (possibly with equality in one criterion). In some cases, a treatment will be interesting for further research if it is superior in at least one of two criterions, regardless of a possible inferiority in the other criterion. This corresponds to the three-quadrant alternative (cf. Figure 2.2 (d)). We could think of much more general forms of dividing up the plane. However, the sectors have the advantage of remaining unchanged if both coordinate axes are scaled by the same positive factor. The class of sector alternatives is even closed with respect to affine transformations. From a practical point of view, especially in a context where nonparametric tests are desirable, it is also often difficult to specify more than the directions that we expect for a change of the location parameter. For the case of p > 2 dimensions, several authors (e. g. Perlman, 1969; Silvapulle, Silvapulle, and Basawa, 2002) have considered positively homogeneous sets or cones (in a very general sense of the word) as alternative regions, i. e. sets C R p with the property that x C implies ax C for all positive real numbers a. The most widely-used positively homogeneous sets are the convex cones. However, neither of these terms is an exact generalization of the concept of sectors to the multivariate case: In R 2, a sector with an angle of more than π is not a convex cone, while e. g. any union of two sectors is a positively homogeneous set but not necessarily a sector. We can define a true generalization of a sector

14 6 2 One-Sided Multivariate Location Hypotheses u 2 1 ϕ 2 1 ϕ (a) 1 u 1 (b) (c) (d) 1 1 Figure 2.2: (a) Specification of a sector alternative; (b) Special sector alternative with ϕ 2 = ϕ 1 +π, i. e. half-plane; (c) Special sector alternative with ϕ 1 = 0,ϕ 2 = π 2, i. e. first quadrant; (d) Special sector alternative with ϕ 1 = π 2,ϕ 2 = π, i. e. first, second, and fourth quadrant.

15 2.2 Simple vs. Composite Null Hypotheses 7 by requiring a positively homogeneous set C R p to have a simply connected intersection with the unit sphere S p 1. We call such a set C an s-simply-connected cone. In many cases, it may be sensible to restrict attention to polyhedral cones, which can be specified by linear inequalities. Again, an important special case for hypotheses based on componentwise criteria are the orthants, which are the multivariate analogues of the quadrants. In Section 2.3, we will deal with some modifications of the one-orthant alternative that do not fit into the framework of this section. 2.2 Simple vs. Composite Null Hypotheses In the preceding section, we have written the null hypothesis as H 0 : ϑ = 0, i. e. as a simple (point) hypothesis. In the univariate case, for the tests widely used, it is e. g. irrelevant whether H 0 : ϑ = 0 or H 0 : ϑ 0 is tested against H 1 : ϑ > 0: If the test keeps its level for H 0 : ϑ = 0, it also keeps its level for the composite null hypothesis. In the multivariate case, however, such a difference may be important, and, depending on the context, a composite null hypothesis may be more appropriate. Let H 1 : ϑ Θ 1 be the multivariate restricted alternative considered, e. g. with a positively homogeneous set for Θ 1. If we only consider a restricted alternative in order to enhance the power for ϑ Θ 1 (because of some a priori conjecture) and if the aim is to show that ϑ is different from 0, we can use the simple null hypothesis ϑ = 0. For ϑ R p ({0} Θ 1 ), we do not have to consider the level or the power in this case. The same trivially applies if the set Θ R p of theoretically possible parameter values does not contain any point outside of {0} Θ 1. If, in contrast, {0} Θ 1 is a true subset of Θ and if we want to show a deviation into a specific direction (the region Θ 1 ), we have to consider a rejection of the null hypothesis for ϑ Θ ({0} Θ 1 ) as a type I error. In this case, tests with unrestricted alternatives do not only have poor power, they even do not necessarily respect the specified level. The appropriate hypotheses are then H 0 : ϑ Θ Θ 1 vs. H 1 : ϑ Θ 1, i. e. the null hypothesis is composite. A typical example is that an amelioration (in some sense) in a treatment group with respect to a control group has to be shown. Berger (1982) mentions the example of a product that should be shown to meet all of several standards. In such cases, a composite null hypothesis is needed. By composite null hypothesis, we will usually mean the largest possible composite null hypothesis, i. e. H 0 : ϑ Θ 0 = Θ Θ 1 ; intermediate cases will be indicated explicitly. Often, Θ will be taken to be R p for simplicity. The case where H 0 is composite is more appropriate for many applications, but also more difficult to handle than the case of a simple null hypothesis because the level has to be met for every ϑ Θ 0 = Θ Θ 1. (This may be the reason why

16 8 2 One-Sided Multivariate Location Hypotheses (a) ε 1 ε 2 u 2 ϕ 2 ε 1 ϕ 1 ε 2 (b) u1 Figure 2.3: Special alternative hypotheses for non-inferiority problems: (a) Alternative region as in Bloch, Lai, and Tubert-Bitter (2001); (b) Intersection with a sector alternative. it has only rarely been treated in the literature.) We will consider both types of null hypotheses occasionally. Namely, while we will use simple null hypotheses for the construction of some of our tests, we will place emphasis on the case of composite null hypotheses for assessing the performance of the tests. One of the few articles that emphasize the importance of the distinction between the two types of null hypotheses is the one by Tang (1998). 2.3 Hypotheses for Non-Inferiority Problems In univariate non-inferiority problems (also called one-sided equivalence problems ), the intention is to show that some parameter ϑ is above or at least approximately equal to a specified value ϑ 0. Such a problem can be formalized as H 0 : ϑ ϑ 0 ε against H 1 : ϑ > ϑ 0 ε, where ε > 0 is some deviation that is considered irrelevant (e. g. Wellek, 2003). This is just a shifted version of the usual one-sided problem. A more sophisticated formulation is based on an indifference region (Jennison and Turnbull, 1993) or range of equivalence (Freedman, Lowe, and Macaskill, 1983) between ϑ 0 ε and ϑ 0 +δ that separates the null and alternative parameter regions. If the true parameter is within this interval, it is not important whether we decide for H 0 or for H 1, and therefore we can neglect the rejection probabilities in this interval. When we turn to the multivariate case, various approaches are possible: Bloch, Lai, and Tubert-Bitter (2001) propose to test H 0 : ϑ R p Θ 1 against H 1 : ϑ Θ 1 with Θ 1 = {ϑ : ϑ j > ε j, j 1,...,p, and j 1,...,p : ϑ j > 0}. This alternative region Θ 1 is shown in Figure 2.3 (a) for the bivariate situation. Jennison and Turnbull (1993) also propose this kind of alternative as one possibility;

17 2.3 Hypotheses for Non-Inferiority Problems 9 δ 2 δ 2 ε 1 δ 1 δ 1 ε 1 (a) ε 2 (b) ε 2 Figure 2.4: Non-inferiority alternative hypotheses (hatched) with indifference regions (dotted): (a) Cartesian product of componentwise indifference regions; (b) Indifference region reduced to the points where not all components have the same sign. the other three possibilities proposed in their Figure 2 are shifted versions of an orthant alternative. Conaway and Petroni (1996) formulate a sector alternative for the vector of the response rate and the toxicity rate in a tumor therapy. They allow for a trade-off between response and toxicity e. g. a toxicity rate slightly above some reference value might be acceptable if the response rate is clearly above its reference value. This is a genuinely bivariate characterization of non-inferiority, in contrast to the ideas based on componentwise non-inferiority. We can also combine the two approaches above: If we take the intersection of a sector alternative and the alternative from Figure 2.3 (a), the resulting alternative region is given in Figure 2.3 (b). It allows for a trade-off between the two parameters near the origin (in the sense proposed by Conaway and Petroni), but for each parameter, a strict lower bound is also given. Based on the univariate indifference region approach, the most obvious multivariate version is the specification of the multivariate indifference region as a Cartesian product ( ε 1,δ 1 ]... ( ε p,δ p ], as in Figure 2.4 (a). It may also be interesting to look at a slightly smaller indifference region that does not include parameter values having the same sign in all components, as shown in Figure 2.4 (b). As can easily be seen, the border between null and alternative region in both parts of Figure 2.3 goes through the indifference region in Figure 2.4 (b) if tanϕ 1 = ε 2 /δ 1 and tan(ϕ 2 π/2) = ε 1 /δ 2. This means that if some level α test for one of the problems without indifference region has some given power for every point in the alternative region, then this test is also of level α (or less) and has (at least) the same power for the situation in Figure 2.4 (b). If, in turn, a test is suitable when using the reduced indifference region from Figure 2.4 (b), it is also suitable for problems with the full indifference region in Figure 2.4 (a).

18 10 2 One-Sided Multivariate Location Hypotheses Tamhane and Logan (2004) propose a similar combination of hypotheses as in Figure 2.4 (a), but they incorporate the indifference region into the null hypothesis. This is just a shifted version of the situation in Figure 2.3 (a). 2.4 Translations of the Hypotheses For notational simplicity, we have formulated all hypotheses in a version that is centered at 0. Of course, we also need tests for any hypothetical location parameter value other than the origin or for composite null and alternative regions that are constructed around some point (a meta-parameter γ R p ) other than the origin, i. e. for translated (shifted) versions of our various hypotheses. We will maintain the simplified setting based on centered hypotheses in the following chapters. While doing so, however, we tacitly assume that tests for ϑ Θ 0 (γ) = γ+θ 0 (0) against ϑ Θ 1 (γ) = γ+θ 1 (0) are derived by application of a test for ϑ Θ 0 (0) against ϑ Θ 1 (0) to the translated data X 1 γ,...,x n γ, for all γ Γ, where Γ R p is the set of possible meta-parameters. We also maintain the short notation Θ 0 = Θ 0 (0) and Θ 1 = Θ 1 (0). For the construction of confidence regions in Chapter 12, we will have to return to the general setting and the notation with a meta-parameter γ. 2.5 Two- and Multi-Sample Problems For the two-sample problem, we can formulate hypotheses analogously to the one-sample case if we consider the difference of the location parameters, ϑ 2 ϑ 1. Similarly, in the multi-sample case, we can consider differences between location parameters of consecutive samples to formulate hypotheses. As in the univariate case, we may postulate a strictly positive difference (in the sense of the one-sample alternative) for at least one such comparison, and a nonnegative difference (i. e. a zero difference is also allowed) for the others. There is an additional one-sided problem that has no analogue in the unrestricted context: the test for a monotone trend in a sequence of random vectors. This can be regarded as a degenerate multi-sample problem where each sample consists of one observation only.

19 11 Chapter 3 Properties of Tests 3.1 Invariance Properties In the univariate setting, the common nonparametric tests are invariant with respect to quite general classes of transformations of the data. E.g., tests based on the ranking of the data are obviously invariant under strictly increasing transformations. For unrestricted alternatives, there has been a variety of proposals for multivariate nonparametric location tests having desirable invariance properties under certain transformations; see e. g. Oja (1999) for a review of affine invariant methods. We give an overview of some possible invariance properties and discuss their relevance in the context of one-sided tests Affine Transformations of p-variate Data Definition Let x be a p-variate (observation) vector, A a p p matrix, and b a (fixed) p-variate vector. (a) x x + b is a location transformation or translation. (b) x Ax is a scale transformation if A is nonsingular and diagonal, i. e. a diagonal matrix with non-zero diagonal elements. (c) x Ax is a component permutation if each row and each column of A contains exactly one 1 and all other entries are 0. (d) x Ax is a rotation if A is orthogonal with determinant 1. (e) x Ax is an orthogonal transformation if A is orthogonal. (f) x Ax + b is an affine transformation if A is nonsingular. Obviously, any location and/or scale transformation is affine. Any component permutation is an orthogonal transformation, any rotation is also an orthogonal

20 12 3 Properties of Tests transformation, and any orthogonal transformation is affine. Thus the class of affine transformations contains all the transformations in the above definition. If n p-variate observations are written as row vectors of an n p matrix X and each observation is transformed according to x Ax + b, this can be written as X XA T + B, where B is an n p matrix containing b T in each row, and the result is again an n p matrix with the transformed observations in its rows Invariance of Tests Definition Let T be a class of transformations T : R p R p. Let S Θ0,Θ 1 be a test statistic for H 0 : ϑ Θ 0 vs. H 1 : ϑ Θ 1, based on n p-variate observations X 1,...,X n, with Θ 0 R p, Θ 1 R p Θ 0. S is T -invariant if, for all T T and X i R p,i = 1,...,n, S Θ0,Θ 1 (X 1,...,X n ) = S T(Θ0 ),T(Θ 1 )(T(X 1 ),...,T(X n )). Application of this definition to the transformation classes in Definition leads to the concepts of translation invariance, scale invariance, component permutation invariance, rotation invariance, orthogonal invariance, and affine invariance, respectively More General Transformation Classes In the univariate case, invariance of tests can not only be postulated for affine (i. e. linear) transformations. Therefore, in the generalization to the multivariate case, we can try to preserve invariance under more general classes of transformations than the affine ones. Of course, we can consider componentwise strictly monotone transformations, which can be approximated by application of bicontinuous transformations to each component (componentwise homeomorphisms). As a class of genuinely multivariate transformations resembling the univariate concept of strict monotonicity, bicontinuous transformations (homeomorphisms) can also be considered. Figure 3.1 gives an overview of the inclusions between some of the transformation classes discussed here and in Definition Transformations and One-Sided Tests Under affine transformations, rays are mapped to rays, and positively homogeneous sets are mapped to positively homogeneous sets if the origin is moved appropriately. Therefore, it is adequate to ask whether a test for a positively homogeneous alternative is invariant under affine transformations (or some subclass of affine transformations). Similar considerations apply for componentwise strictly monotone transformations in the context of orthant alternatives. If we

21 3.1 Invariance Properties 13 translations scale transformations location & scale transformations componentwise homeomorphisms component permutations affine transformations orthogonal transformations rotations homeomorphisms Figure 3.1: Inclusions between the different classes of transformations of multivariate data. require affine invariance and invariance under componentwise strictly monotone transformations, we reach a high degree of independence from the way the data is represented in the variables. However, these invariance requirements substantially constrain the choice of tests, and they are unnecessary for many multivariate applications. Namely, we can use less stringent requirements if we restrict our attention to special (but frequent) types of data and hypotheses: For a direction alternative, a half-space is associated with the hypothetical direction, the border being the hyperplane that is orthogonal to the given direction. This orthogonality property is generally not preserved under affine (or scale) transformations, and therefore, invariance under such transformations may not be a sensible requirement. (Nevertheless, such a requirement may be sensible if the orthogonality, i. e. the scalar product, is defined in a coordinate system that depends on the distribution of the data.) Orthogonal transformations do not change orthogonality, and tests for direction alternatives may sensibly be investigated for the respective invariance properties. If we examine genuinely spatial data (i. e. data with all variables measured in the same units and with an arbitrary orientation of the coordinate system), orthogonal invariance is an obvious requirement, while e. g. scale changes are only sensible if they are applied to all variables simultaneously. Alternatives that are defined using componentwise criteria (such as orthant alternatives and most of the alternatives from Section 2.3) do not usually stay within their class under more general transformations than componentwise homeomorphisms. Such alternatives are most interesting in the case of data that is not

22 14 3 Properties of Tests genuinely spatial, where the variables represent an arbitrary collection of measurements typically measured in different units. But in this case, there is hardly any justification for applying rotations or even general affine transformations. By construction (see Section 2.4), the tests proposed in the following chapters will be translation invariant. Most sensible tests will also be invariant under permutations of the components. Note that even a test for hypotheses with different criteria for the different components may well be invariant under permutations of the components since the transformation is also applied to the hypotheses for assessing invariance. A situation where this invariance property is not required could be the case of several univariate hypotheses of different importance. The approach using bicontinuous transformations does not seem to be promising for one-sided problems: It is not difficult to find a bicontinuous transformation mapping two points that are in the same direction from the origin to points that are in opposite directions, while leaving the origin unchanged. (Imagine e. g. a bicontinuous transformation T : R 2 R 2 with T((0, 0) T ) = (0, 0) T, T((1, 0) T ) = (1, 0) T and T((2, 0) T ) = ( 2, 0) T.) Thus, it is not appropriate to require a test to be invariant under such transformations if one-sided hypotheses formulated in terms of directions or positively homogeneous sets are considered. 3.2 Unbiasedness In univariate problems, a test is usually required to be unbiased, i. e. its power function should be at most α for ϑ Θ 0 and at least α for ϑ Θ 1. Lehmann (1952) investigates the situation where H 0 : ϑ i 0,i = 1,...,p is tested against H 1 : i : ϑ i > 0. He shows that the only test that has an analytic power function and that is unbiased for this problem is the trivial test with constant power α. We generalize this result to the general sector alternative (excluding only the half-space case) with the full composite null hypothesis. We do this in two steps: Corollary Let a sample (X 1,...,X n ) from a distribution from some family (P ϑ ) ϑ R 2 be given. Let ϕ : (R 2 ) n [0, 1] be a test for H 0 : ϑ Θ 0 vs. H 1 : ϑ Θ 1, where Θ 1 = R 2 Θ 0 is a sector with angle ψ > π. Assume that ϕ is an unbiased level α test and that it has an analytic power function. Then the test is trivial, i.e. the power of the test is constant: β(ϑ) = α under all ϑ R 2. Proof. Let T : R 2 R 2 be an affine transformation such that T(Θ 0 ) is the negative quadrant. Define transformed observations X i = T(X i ), which have the distribution P ϑ = P T 1 ( ϑ) T 1. ϕ( X 1,..., X n ) = ϕ(t 1 ( X 1 ),...,T 1 ( X n )) can be used to test H 0 : ϑ Θ 0 vs. H1 : ϑ Θ 1 with Θ j = T(Θ j ),j = 0, 1. If β denotes the power function of ϕ, then β(t(ϑ)) = β(ϑ).

23 3.3 Cone Order Monotonicity 15 For a sample from a distribution from the family ( P ϑ) ϑ R 2, ϕ is therefore also an unbiased level α test for H 0 vs. H1 and has an analytic power function. Because Θ 0 = T(Θ 0 ) is the negative quadrant, Lehmann s result can be applied, and β( ϑ) = α, ϑ R 2. But this implies also β(ϑ) = α, ϑ R 2. Corollary Let a sample (X 1,...,X n ) from a distribution from some family (P ϑ ) ϑ R 2 be given. Let ϕ : (R 2 ) n [0, 1] be a test for H 0 : ϑ Θ 0 vs. H 1 : ϑ Θ 1, where Θ 1 = R 2 Θ 0 is a sector with angle ψ < π. Assume that ϕ is an unbiased level α test and that it has an analytic power function. Then the test is trivial, i.e. the power of the test is constant: β(ϑ) = α under all ϑ R 2. Proof. Define ϕ(x 1,...,X n ) = 1 ϕ(x 1,...,X n ), and let its power function be β. Trivially, β(ϑ) = 1 β(ϑ). Therefore, ϕ is an unbiased level 1 α test for H 0 : ϑ Θ 1 vs. H 1 : ϑ Θ 0, and its power function is analytic. Due to Corollary 3.2.1, β(ϑ) = 1 α, ϑ R 2, and it follows that β(ϑ) = α, ϑ R 2. From these results, we can see that strict unbiasedness will not be a useful criterion in most bivariate (and presumably also higher dimensional) problems, as long as the null hypothesis is composite. However, it is still desirable to have a nearly unbiased test. One method for assessing the bias of a test will be given in Section Cone Order Monotonicity Cohen and Sackrowitz (1998) propose that tests for cone alternatives should satisfy the cone order monotonicity property with respect to Θ 1 and/or its positive dual Θ 1 = {a : a T ϑ 0 ϑ Θ 1 }. Their definition of cone order monotonicity only applies to a test based on a single p-variate statistic. When we look at a general test based on a sample of n p-variate observations, at least two generalizations are possible; we introduce these in parts (b) and (c) of the following definition. Definition Let C R p be a convex cone. (a) A function f : R p R is cone order monotone with respect to C if f(x) f(x + δ) x R p,δ C. (b) A function f : (R p ) n R is cone order monotone in the sample with respect to C if f(x 1,...,x n ) f(x 1 + δ,...,x n + δ) x 1,...,x n R p,δ C.

24 16 3 Properties of Tests (c) A function f : (R p ) n R is cone order monotone in each observation with respect to C if f(x 1,...,x n ) f(x 1 + δ 1,...,x n + δ n ) x 1,...,x n R p, δ 1,...,δ n C. It is obvious that cone order monotonicity in each observation implies cone order monotonicity in the sample. For a discussion of the adequacy of requiring cone order monotonicity, see e. g. Perlman and Chaudhuri (2004) and Cohen and Sackrowitz (2004). Cohen and Sackrowitz (1998) give a method for constructing cone order monotone tests from tests for unrestricted alternatives by enlarging the acceptance region (and then determining the level of the new test). The authors always use the smallest superset of the acceptance region such that the monotonicity condition is fulfilled. However, one could also use a larger superset of the same acceptance region, which would lead to a test with even smaller level. The resulting family of tests might be more appropriate for certain composite null hypotheses. 3.4 Assumptions on the Underlying Distribution For a test to be valid, the distribution of the observations has to belong to a certain parametric family in the case of parametric tests. For nonparametric one-sample tests, we typically have to assume some kind of symmetry of the distribution. Multivariate distributions can belong to several symmetry classes: Definition Let X be a p-variate random vector. The distribution F X of X is (a) spherically symmetric (with respect to 0) if, for every orthogonal p p matrix A, X d = AX; (b) in the spherical directions class (with respect to 0) if there exists a random vector Y with a spherically symmetric distribution (with respect to 0) such X d that = Y ; X Y (c) elliptically symmetric (with respect to 0) if there exists a nonsingular p p matrix B such that the distribution of BX is spherically symmetric; (d) in the elliptical directions class (with respect to 0) if there exists a random vector Y with an elliptically symmetric distribution (with respect to 0) X d such that = Y ; X Y

25 3.4 Assumptions on the Underlying Distribution 17 N p (,σ 2 I p ) = = spherical symmetry multivariate normality = = = spherical directions elliptical symmetry = = = elliptical directions diagonal symmetry = = directional symmetry Figure 3.2: Implications between the different types of symmetry of multivariate distributions. (e) diagonally symmetric (with respect to 0) if X d = X; (f) directionally symmetric (with respect to 0) if X X d = X X. For ϑ R p, F X is symmetric (in any of the above senses) with respect to ϑ if X ϑ is symmetric (in the same sense) with respect to 0. Spherical symmetry trivially implies the spherical directions property (with Y = X) and elliptical symmetry (with the identity matrix I p for B). As a consequence of the latter implication, a distribution in the spherical directions class is also in the elliptical directions class. Elliptical symmetry implies the elliptical directions property (with Y = X) and diagonal symmetry (with A = I p BX d = BX d = B( X) X d = X). Diagonal symmetry and the elliptical directions property each again trivially imply directional symmetry. These relationships and the obvious relationships with the special case of multivariate normal distributions are illustrated in Figure 3.2, which is an extension of Figure 1 in Randles (2000). If X has a probability density function f X, we can also characterize some of the above symmetry properties as follows: Spherical symmetry: f X (x) = g( x ) for some g : [0, ) [0, ). Elliptical symmetry: f X (x) = g( x T B T Bx) det B, or, in a more habitual form, f X (x) = g(x T Σ 1 x) det Σ 1 2, with Σ = (B T B) 1.

26 18 3 Properties of Tests Diagonal symmetry: f X (x) = f X ( x) for all x R p. Diagonal symmetry is often just called symmetry. Further, diagonal symmetry is also known as central symmetry or reflected symmetry, and directional symmetry is also called angular symmetry ; see e. g. Small (1990), Neuhaus and Zhu (1999). The terms used here are mainly based on those used in Hettmansperger and McKean (1998) and Randles (2000). Randles s definition of the elliptical directions class for observations X 1,...,X n is slightly less restrictive: He does not require the observations to be i. i. d., while we maintain the classical framework of a test based on an i. i. d. sample. The following definition by Neuhaus and Zhu (1999) is equivalent to the one given here: The distribution of X is in the elliptical directions class (with respect to 0) if there exists a matrix B such that BX is uniformly distributed on the unit sphere. BX We have introduced the spherical directions class here in analogy to the elliptical directions class, and it is easily seen that an equivalent (and simpler) X definition would be that is uniformly distributed on the unit sphere. X

27 19 Chapter 4 One-Sided Tests from the Literature There has been a variety of proposals for tests with one-sided, ordered, or restricted alternatives in multivariate settings. The emphasis in this survey is on nonparametric methods. As the hypotheses for multivariate location tests with restricted alternatives are often formulated in terms of componentwise comparisons, the following notation will be useful, given two vectors a = (a 1,...,a p ) T and b = (b 1,...,b p ) T : a b a i b i, i = 1,...,p a b a i b i, i = 1,...,p, and i : a i < b i a < b a i < b i, i = 1,...,p a b, a b, and a > b will be used analogously. 4.1 One-Sample Problem Brown (1983) describes an angle test, a rotation invariant bivariate analogue of a sign test based on the statistic cos(ψ i ϕ), i where ψ i is the angle between the positive part of the first coordinate axis and the i-th observation, and ϕ is the hypothetical direction as in Figure 2.1. Brown uses a normal approximation to the distribution of the test statistic under the null hypothesis. In the introduction to their article, Larocque and Labarre (2004) announce to propose a test for the one-orthant alternative H 1 : ϑ 0. However, the

28 20 4 One-Sided Tests from the Literature conditionally distribution-free sign test proposed actually seems to be more appropriate for the alternative that at least one component of ϑ is positive, without any restriction being imposed on the other components (bivariate case: three-quadrant alternative, Figure 2.2 (d)). The test is based on the supremum of univariate sign test statistics on projections of the data and will be presented in more detail in Section 9.2. Chinchilli and Sen (1981) give a test of H 0 : ϑ = (ϑ 1,...,ϑ p ) T = 0 against H 1 : ϑ 0, ϑ i 0 for i = 1,...,a, with a p fixed, where ϑ is the parameter vector of a general linear model. They use the union-intersection principle (see Section 5.3). In the special case a = p = 2, the alternative hypothesis corresponds to Figure 2.2 (c). Silvapulle, Silvapulle, and Basawa (2002) develop a class of adaptive tests for the null hypothesis ϑ = 0 against the alternative ϑ C {0}, where ϑ R p is a parameter vector and C is a closed convex subset of R p containing the origin (or, as a special case, a closed convex positively homogeneous set). This setting includes the half-plane and one-quadrant alternatives discussed for the bivariate case (Figure 2.2 (b), (c)). The approach of these authors is based on asymptotic considerations. Minhajuddin, Frawley, Schucany, and Woodward (2006) propose two bootstrap tests. The first one is for the simple null hypothesis ϑ = 0 against the alternative that ϑ is in the positive orthant. It is based on the likelihood ratio test statistic used by Perlman (1969) in the multivariate normal context see below. The second test proposed is for the same alternative, but for the composite null hypothesis consisting of the complement of the positive orthant. Resampling is done after centering the data in such a way that the mean is on the border of the null parameter region whenever the original mean was in the positive orthant. Parametric procedures: Kudô (1963) considers a p-variate normal population with mean ϑ and known covariance matrix. He proposes a likelihood ratio test for H 0 : ϑ = 0 against H 1 : ϑ 0. This alternative hypothesis corresponds to the bivariate situation in Figure 2.2 (c). Perlman (1969) extends this work to positively homogeneous sets as alternative regions and to the case of an unknown covariance matrix, giving upper and lower bounds on the null distribution of the test statistic. A discussion of these tests and related results is given in Section 4.6 of Robertson, Wright, and Dykstra (1988). An approximation to these likelihood ratio tests is given by Tang, Gnecco, and Geller (1989).

29 4.1 One-Sample Problem 21 For a summary of early parametric approaches in the context of (convex) positively homogeneous sets, see Shapiro (1988). Akkerboom (1990) proposes to test a simple null hypothesis against a polyhedral cone alternative by using a circular cone that approximates the original polyhedral cone. A composite null hypothesis problem is investigated by Sasabuchi (1988). He considers a p-variate normal random vector with mean ϑ and unknown covariance matrix. For given p-variate vectors b 1,...,b k satisfying certain conditions, he derives the likelihood ratio test of H 0 : b T i ϑ 0 for all i with at least one equality against H 1 : b T i ϑ > 0 for all i. In this context, the above formulation of the null hypothesis is equivalent to H 0 : i : b T i ϑ 0. See also Berger (1989), where two more powerful modifications of the test are proposed, of which at least one has counterintuitive properties. Further similar modifications are proposed by McDermott and Wang (2002), assuming a known covariance matrix. In order to test if the components have positive means (assuming a known covariance matrix Σ), Follmann (1996) proposes to use the likelihood ratio test statistic X 2 = n X T Σ 1 X, but to reject the null hypothesis if both X 2 exceeds the critical value for the level 2α and the sum of the componentwise means is positive. In the case of an unknown Σ, Hotelling s T 2 can be used instead of X 2. In the bivariate case, the alternative is similar to the one in Figure 2.2 (b) (with ϕ 1 = π/4) or Figure 2.1 (with ϕ = π/4). Cohen and Sackrowitz (1998) give a general procedure to derive tests for H 0 : ϑ = 0 vs. H 1 : ϑ C {0} (where C is a closed convex positively homogeneous set) from tests for unrestricted alternatives when ϑ is the natural parameter of an exponential family. Their procedure enlarges the acceptance regions of unrestricted tests in order to ensure cone order monotonicity of the tests see Section 3.3. The significance levels for these enlarged acceptance regions are not directly related to those of the unrestricted tests and therefore have to be determined from scratch. Mudholkar, Kost, and Subbaiah (2001) propose a robustified test based on trimmed means. Glimm, Srivastava, and Läuter (2002) test H 0 : ϑ = 0 against H 1 : ϑ 0. They give the exact null distribution of several versions of a test statistic under a normal distribution. In order to simplify the calculation of these test statistics, the convex polyhedral cone resulting from an affine transformation of the positive orthant is approximated by an orthant. According to the authors, it can be shown that the null distribution of the test statistics is valid for data from any elliptically symmetric distributions. For the problem of showing that at least one component of the parameter is

30 22 4 One-Sided Tests from the Literature positive, Perlman and Wu (2006) propose a test that is more powerful than the likelihood ratio and union-intersection tests in cases where some of the components are negative. They also provide a related test for showing that at least one component is positive while the other components are above some slightly negative value, as in Figure 2.3 (a) for the bivariate case. A survey of other, mostly parametric approaches can be found in Sen and Silvapulle (2002). 4.2 Two-Sample Problem Bhattacharyya and Johnson (1970) propose a layer rank test for the bivariate case. They consider two independent random samples (Z 1,...,Z n1 ) and (Z n1 +1,...,Z n1 +n 2 ), where Z i = (X i,y i ) T follows a continuous distribution F for i = 1,...,n 1 and G for i = n 1 + 1,...,n 1 + n 2, respectively. The test problem is H 0 : F G vs. H 1 : F G,F(x,y) G(x,y), F(x,y) Ḡ(x,y) for all (x,y), where F(x,y) := P(X 1 x,y 1 y). (Under the conditions of H 1, a random vector with cdf F is called strongly stochastically smaller than a random vector with cdf G.) Thus Bhattacharyya and Johnson discuss a more general location problem than the shift problem presented in Section 2.1. The test statistic used is [ 1 n 1 (n 1 + n 2 ) 2 n 1 + n 2 n 2 n 1 + n 2 n 1 +n 2 n 1 +n 2 i=n 1 +1 j=1 n 1 n 1 +n 2 i=1 j=1 1(X i X j,y i Y j ) 1(X i X j,y i Y j ) on which a permutation test is performed. It is shown that the statistic is invariant under the group of bicontinuous transformations g : R 2 R 2 satisfying z 1 z 2 g(z 1 ) g(z 2 ), where the inequalities are to be interpreted componentwise, as defined in the introduction to this chapter. Johnson and Mehrotra (1972) compare this test statistic to a more elaborate one, also based on coordinatewise rankings. Wei and Knuiman (1987) adapt the test to the situation with censored data. ],